After seeing all these, I went back to my breadboard, I setup a 20KHz PWM using 555 timer, took my transformer and connected the 7v side to my mosfet bridge, I connected the 220v side to mains AC, without switching, I got 4v DC from mosfet bridge, I powered the breadboard and start switching, at first, no voltage increase, so, I started increasing the duty cycle starting from 10% till I got to 50%, and the voltage gradually increased from 4v to 5v, which is not what I was expecting.

I tried everything I could, I couldn't get pass 5v DC, i bumped up the frequency, still no difference, so I decided to change the output capacitor from 35v 100uF to 35v 1000uF, and I was able to get to 5.5v.

Then I changed it again to 16v 3300uF and I was able to get to 6.3v, so I added 3 of the 3300uF capacitors in parallel, but no difference. So, my best result was 6.3v

I don't know what is happening or what I am doing wrong.

Your experiments have been most helpful. According to me, the fact that you have gotten this results, you have already succeeded: here is why...

In all the UPS I have dissected (>40 and >5 different brands all of them for 12v batteries), the transformer always have at least one extra input coil on the primary winding. The extra coil when fed to mains produces a voltage of about ~11vac while the one used by the inverter produces ~7vac. A relay is usually attached on the hot side to switch between the 2 input wires which are often from the same coil. The 11v coil when rectified should give >13.8vdc which in my thinking is sufficient assuming no boost regulation. With your 50% increase (4v to 6.3v) that would give you close to 20v offering you enough room to pwm for whatever current charging needs.

 

I do not see that I ever got a response as to if this was for a commercial product design, or a one-off personal project.
If it were my own personal project I would have one portion being the inverter supply to deliver a backup mains voltage, a change-over circuit for the UPS to engage the load, and a totally separate battery charger to provide a reasonable recovery time as well as a float charge output for maximum battery life. None of the portions being optimized for minimum cost and maximum profit. The "best" designs are seldom the cheapest ones or the simplest ones.